2003 Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers
Reinhard Farwig
Adv. Differential Equations 8(3): 357-384 (2003). DOI: 10.57262/ade/1355926858

Abstract

The aim of this paper is to present a new joint approach to the Helmholtz decomposition in infinite cylinders and in infinite layers $\Omega=\Sigma {\times} {\mathbb R}^k$ in the function space $L^q({\mathbb R}^k;L^r({\Sigma}))$ using even arbitrary Muckenhoupt weights with respect to $x'\in \Sigma\subset {\mathbb R}^{n-k}$ and, if possible, exponential weights with respect to $x''\in {\mathbb R}^k, \;1\leq k\leq n-1,\;n\geq 2.$ For $n=2$, we get the Helmholtz decomposition for a strip, for $n=3$ in an infinite cylinder or an infinite layer and for $n>3$ in some (non--physical) unbounded domains of cylinder or layer type. The proof based on a weak Neumann problem uses a partial Fourier transform and operator--valued multiplier functions, the ${\mathcal R}$--boundedness of the family of multiplier operators and an extrapolation property in weighted $L^q$--spaces.

Citation

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Reinhard Farwig. "Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers." Adv. Differential Equations 8 (3) 357 - 384, 2003. https://doi.org/10.57262/ade/1355926858

Information

Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1038.35068
MathSciNet: MR1948530
Digital Object Identifier: 10.57262/ade/1355926858

Subjects:
Primary: 35Q30
Secondary: 35J20 , 35J25 , 76D05

Rights: Copyright © 2003 Khayyam Publishing, Inc.

Vol.8 • No. 3 • 2003
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